The Veneziano amplitude in AdS$_5 \times$S$^3$ from an 8-dimensional effective action

We study four-point functions of arbitrary half-BPS operators in a 4-dimensional $\mathcal{N}=2$ SCFT with flavour group $SO(8)$ at genus-zero and strong 't Hooft coupling, corresponding - via AdS/CFT - to the ($\alpha'$ expansion of the) Veneziano amplitude on an AdS$_5 \times$S$^3$ background. We adapt a procedure first proposed by Abl, Heslop and Lipstein in the context of AdS$_5 \times$S$^5$, and postulate the existence of an effective action in terms of an $8$-dimensional scalar field valued in the adjoint of the flavour group. The various Kaluza-Klein correlators can then be computed by uplifting the standard AdS/CFT prescription to the full product geometry with AdS bulk-to-boundary propagators and Witten diagrams replaced by suitable AdS$_5 \times$S$^3$ versions. After elucidating the main features of the procedure, valid at all orders in $\alpha'$, we show explicit results up to order $\alpha'^{5}$. The results provide further evidence of a novel relation between AdS$\times$S and flat amplitudes - which made its first appearance in $\mathcal{N}=4$ SYM - that is perhaps the most natural extension of the well known flat-space limit proposed by Penedones to cases where AdS and S have the same radius.